Lacroix and the Calculus pp 89-139 | Cite as

# Analytic and differential geometry

Chapter

## Abstract

The two final chapters in volume I comprise a “complete theory of curves and curved surfaces”; that is, not only the “application of the differential calculus to the theory of curves” (and of curved surfaces) — what we now call differential geometry — but also the “purely algebraic part of that theory” — analytic geometry. Lacroix explained the inclusion of analytic geometry by his desire to offer a full set (“ensemble complet”) and to relate notions that were usually presented from very different points of view [*Traité*, I, xxv, 327].

## Keywords

Tangent Plane Descriptive Geometry Algebraic Curf Plane Curf Conic Section
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## References

- 2.Belhoste [
*1992*, 568] reads here “avec vous”; but given the teacher-pupil tone of the rest of the letter, this does not sound very convincing (unless of course Lacroix was writing that separate work as lectures for this student). Belhoste also interprets this whole passage as meaning that Lacroix intended to interpose his “descriptive geometry” [Lacroix*1795*] in the*Traité*. I disagree: Lacroix certainly made many references to [Lacroix*1795*] in chapter 5, but what he says here is that a work he had been writing on the application of analysis (to geometry, presumably) was going to be interposed in the*Traité*— that separate work must correspond to chapters 4 and 5.Google Scholar - 4.In the 16th century the possibility of access to ancient Greek mathematical works had increased considerably because of the printing of both original versions and (usually Latin) translations. This (particularly the publication in 1588 of Commandino’s Latin translation of Pappos’
*Mathematical Collection*) had given origin to what Bos calls “the early modern tradition of geometrical problem solving” [Bos*2001*, ch. 4].Google Scholar - 5.“I choose a straight line, as
*AB*, to which to refer all its points [i.e. those of the curve*EC*], and in*AB*I choose a point*A*at which to begin the investigation. [...] Then I take on the curve an arbitrary point, as*C*, at which we will suppose the instrument applied to describe the curve. Then I draw through*C*the line*CB*parallel to*GA*. Since*CB*and*BA*are unknown and indeterminate quantities, I shall call one of them*y*and the other*x*. [...] the required equation is*y*^{2}=*cy*−_{b}^{cx y}+*ay*−*ac*.” [Descartes*Géométrie*, 51–52]Google Scholar - 6.But it should be mentioned that several of Fermat’s works, including the
*Isagoge*, had circulated much before, in manuscript form, among the Parisian mathematicians [Boyer*1956*, 82; Bos*2001*, 205−206].Google Scholar - 8.Quoted in [Boyer
*1956*, 148].Google Scholar - 10.For this claim, see [Boyer
*1956*, 168–170].Google Scholar - 11.According to [Boyer
*1956*, 272] its “treatment of analytic geometry is typical of the time about 1775”.Google Scholar - 12.Lacroix had published a textbook
*Traité élémentaire de trigonométie et d’application de l’algèbre à la géométrie*[*1798b*], combining in one volume these two subjects; [Lacroix & Bézout*1826*] was a combined translation of Lacroix’s trigonometry and Bézout’s application of algebra to geometry.Google Scholar - 14.[Lacroix & Bézout
*1826*] closes just after the study of the conic sections, so that it does not include the construction of equations. It is unlikely that this is due to the obsolescence of the subject, since an 1829 French edition of Bézout’s*Cours*(Paris: Bachelier) still includes that section.Google Scholar - 21.This included what according to Boyer [
*1956*, 205–206] was perhaps the first explicit appearance of the point-slope equation of the straight line:*y*−*y*′ =*a*(*x*−*x*′), where*a*is the tangent of the angle between the straight line and the abscissa axis and*x*′,*y*′ are the coordinates of a given point on it [Monge*1781*, 669].Google Scholar - 22.An abridged syllabus of this course is in [Langins
*1987a*, 130–131]. Of course, there is no guarantee that Monge really followed this syllabus. One serious possibility is that he may have taught only the geometrical applications, while others (Hachette, Malus, Dupuis) taught algebra and the calculus [Langins*1987a*, 78]. See also section 8.2.Google Scholar - 24.“In carefully avoiding all geometric constructions, I would have the reader realize that there exists a way of looking at geometry which one might call
*analytic geometry*, and which consists in deducing the properties of extension from the smallest possible number of principles by purely analytic methods, as Lagrange has done in his mechanics with regard to the properties of equilibrium and movement”. This translation is taken from [Boyer*1956*, 211].Google Scholar - 29.Both Taton [
*1951*, 135] and Boyer [*1956*, 220] wrongly ascribe this little priority to Jean-Baptiste Biot. Biot published in 1802 a*Traité analytique des courbes et des surfaces du second degré*; he changed the title of this work in the second edition (1805) to*Essai de géométrie analytique, appliqué aux courbes et aux surfaces du second degré*. Boyer had the excuse that he apparently did not see the first edition and assumed it had the same title as the second [Boyer*1956*, 273]; but Taton [*1951*, 132] gave all these (and more) bibliographic details.Google Scholar - 30.“should serve for mutual clarification, corresponding, so to speak, to the text of a book and its translation”. This translation is taken from [Boyer
*1956*, 212].Google Scholar - 32.To be more precise, it will be divisible by
*t*^{n}+1, where*n*is the largest integer by which it would be divisible in general (that is, the multiplicity of that point). This procedure can be found in [Cramer*1750*, 460–464] and [Goudin & du Séjour*1756*, 77–78]. Transformation of coordinates are fundamental tools in these books.Google Scholar - 36.An interesting remark is that between the curve and any of these circles it is impossible to pass another circle [l’Hôpital
*1696*, 73]. It is interesting because Lagrange will use this property as a definition of contact.Google Scholar - 45.Of course some regularity is needed for this argument, namely that
*ϕ*″ and*F*″ be bounded in a neighbourhood of*x*. On a different note, there is a printing error here:_{2}^{i}[*f*″(*x*+*j*) −*F*″(*x*+*j*)] instead of_{2}^{i}[*ϕ*″(*x*+*j*) −*F*″(*x*+*j*)]; this was later corrected (at least in the*Œuvres*printing [Lagrange*Fonctions*, 2nd ed, 187]).Google Scholar - 49.Lagrange had also given these simplest osculating curves, but only as a comment, after having dealt with the general theory [
*Fonctions*, 129–130].Google Scholar - 51.It was common belief in the 18th century that all functions were piecewise monotonic. [Lagrange
*Fonctions*, 155–156] for instance, has a similar assumption (also in a proof that the ordinate is the derivative of the area).Google Scholar - 55.This characterization of envelopes can be seen for instance in [l’Hôpital
*1696*, ch. 8].Google Scholar - 57.Henri Pitot had used the name “curves of double curvature” for space curves in 1724, but it was Clairaut [
*1731*] who established it as standard. It was used throughout the 18th century. Neither Pitot nor Clairaut seemed to have in mind first curvature and torsion when using the name [Struik*1933*, 100–101].Google Scholar - 60.For each point in a curve, the radius of curvature is the radius of an evolute, but for two consecutive points in a space curve, the radii of curvature are radii of
*different*evolutes. In fact, there is an important exception to this rule: when the curve is a line of least or greatest curvature of a surface, its centres of curvature do form an evolute. Monge implicitly reported this in [*1781*, 690], stating that the normals are tangent to that curve, but apparently he never recognized explicitly that it is an*evolute*. Lagrange [*Fonctions*, 183], on the other hand, was quite explicit, and Hachette cited him in a footnote in [Monge & Hachette*1799*, 357].Google Scholar - 67.[Tinseau
*1780a*, 593] has an indication of having been submitted in 1774, but according to Taton [*1951*, 76] the correct date is 7 December 1771. [Tinseau*1780b*] has no date but, also according to Taton [*1951*, 76], appears to be contemporary of the former memoir.Google Scholar - 70.As Taton [
*1951*, 210] puts it, these studies take up a score (“une vingtaine”) of chapters out of about twenty-five (“quelque vingt-cinq”) in the differential part of [Monge*Feuilles*].Google Scholar - 71.This last chapter was absent from the first edition [Taton
*1951*, 219].Google Scholar - 78.Although Monge [
*Feuilles*, n° 7-i] had already used it in this sense, applied to surfaces. Lagrange [*Fonctions*] spoke of “courbes enveloppantes” and “surfaces enveloppantes”.Google Scholar - 80.Which is not correct in general. [Coolidge
*1940*, 136] gives the example of any non-planar curve with prime order.Google Scholar

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